math.square on complex, real part

Percentage Accurate: 93.4% → 100.0%

Time: 4.2s

Alternatives: 3

Speedup: 1.2×

• 44.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ re \cdot re - im \cdot im \end{array} \]

FPCore

(FPCore re_sqr (re im) :precision binary64 (- (* re re) (* im im)))

C

double re_sqr(double re, double im) {
	return (re * re) - (im * im);
}

Fortran

real(8) function re_sqr(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    re_sqr = (re * re) - (im * im)
end function

Java

public static double re_sqr(double re, double im) {
	return (re * re) - (im * im);
}

Python

def re_sqr(re, im):
	return (re * re) - (im * im)

Julia

function re_sqr(re, im)
	return Float64(Float64(re * re) - Float64(im * im))
end

MATLAB

function tmp = re_sqr(re, im)
	tmp = (re * re) - (im * im);
end

Wolfram

re$95$sqr[re_, im_] := N[(N[(re * re), $MachinePrecision] - N[(im * im), $MachinePrecision]), $MachinePrecision]

Initial Program: 93.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ re \cdot re - im \cdot im \end{array} \]

FPCore

(FPCore re_sqr (re im) :precision binary64 (- (* re re) (* im im)))

C

double re_sqr(double re, double im) {
	return (re * re) - (im * im);
}

Fortran

real(8) function re_sqr(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    re_sqr = (re * re) - (im * im)
end function

Java

public static double re_sqr(double re, double im) {
	return (re * re) - (im * im);
}

Python

def re_sqr(re, im):
	return (re * re) - (im * im)

Julia

function re_sqr(re, im)
	return Float64(Float64(re * re) - Float64(im * im))
end

MATLAB

function tmp = re_sqr(re, im)
	tmp = (re * re) - (im * im);
end

Wolfram

re$95$sqr[re_, im_] := N[(N[(re * re), $MachinePrecision] - N[(im * im), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \left(im + re\right) \cdot \left(re - im\right) \end{array} \]

FPCore

(FPCore re_sqr (re im) :precision binary64 (* (+ im re) (- re im)))

C

double re_sqr(double re, double im) {
	return (im + re) * (re - im);
}

Fortran

real(8) function re_sqr(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    re_sqr = (im + re) * (re - im)
end function

Java

public static double re_sqr(double re, double im) {
	return (im + re) * (re - im);
}

Python

def re_sqr(re, im):
	return (im + re) * (re - im)

Julia

function re_sqr(re, im)
	return Float64(Float64(im + re) * Float64(re - im))
end

MATLAB

function tmp = re_sqr(re, im)
	tmp = (im + re) * (re - im);
end

Wolfram

re$95$sqr[re_, im_] := N[(N[(im + re), $MachinePrecision] * N[(re - im), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.7%

   \[re \cdot re - im \cdot im \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{re \cdot re - im \cdot im} \]

   2. lift-*.f64 N/A

      \[\leadsto \color{blue}{re \cdot re} - im \cdot im \]

   3. lift-*.f64 N/A

      \[\leadsto re \cdot re - \color{blue}{im \cdot im} \]

   4. difference-of-squares N/A

      \[\leadsto \color{blue}{\left(re + im\right) \cdot \left(re - im\right)} \]

   5. *-commutative N/A

      \[\leadsto \color{blue}{\left(re - im\right) \cdot \left(re + im\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(re - im\right) \cdot \left(re + im\right)} \]

   7. lower--.f64 N/A

      \[\leadsto \color{blue}{\left(re - im\right)} \cdot \left(re + im\right) \]

   8. +-commutative N/A

      \[\leadsto \left(re - im\right) \cdot \color{blue}{\left(im + re\right)} \]

   9. lower-+.f64 100.0

      \[\leadsto \left(re - im\right) \cdot \color{blue}{\left(im + re\right)} \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\left(re - im\right) \cdot \left(im + re\right)} \]

5. Final simplification 100.0%

   \[\leadsto \left(im + re\right) \cdot \left(re - im\right) \]
6. Add Preprocessing

Alternative 2: 96.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot re - im \cdot im\\ t_1 := \left(-im\right) \cdot im\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-312}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;re \cdot re\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore re_sqr (re im)
 :precision binary64
 (let* ((t_0 (- (* re re) (* im im))) (t_1 (* (- im) im)))
   (if (<= t_0 -5e-312) t_1 (if (<= t_0 INFINITY) (* re re) t_1))))

C

double re_sqr(double re, double im) {
	double t_0 = (re * re) - (im * im);
	double t_1 = -im * im;
	double tmp;
	if (t_0 <= -5e-312) {
		tmp = t_1;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = re * re;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double re_sqr(double re, double im) {
	double t_0 = (re * re) - (im * im);
	double t_1 = -im * im;
	double tmp;
	if (t_0 <= -5e-312) {
		tmp = t_1;
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = re * re;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def re_sqr(re, im):
	t_0 = (re * re) - (im * im)
	t_1 = -im * im
	tmp = 0
	if t_0 <= -5e-312:
		tmp = t_1
	elif t_0 <= math.inf:
		tmp = re * re
	else:
		tmp = t_1
	return tmp

Julia

function re_sqr(re, im)
	t_0 = Float64(Float64(re * re) - Float64(im * im))
	t_1 = Float64(Float64(-im) * im)
	tmp = 0.0
	if (t_0 <= -5e-312)
		tmp = t_1;
	elseif (t_0 <= Inf)
		tmp = Float64(re * re);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = re_sqr(re, im)
	t_0 = (re * re) - (im * im);
	t_1 = -im * im;
	tmp = 0.0;
	if (t_0 <= -5e-312)
		tmp = t_1;
	elseif (t_0 <= Inf)
		tmp = re * re;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

re$95$sqr[re_, im_] := Block[{t$95$0 = N[(N[(re * re), $MachinePrecision] - N[(im * im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-im) * im), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-312], t$95$1, If[LessEqual[t$95$0, Infinity], N[(re * re), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 re re) (*.f64 im im)) < -5.0000000000022e-312 or +inf.0 < (-.f64 (*.f64 re re) (*.f64 im im))

   1. Initial program 91.1%

      \[re \cdot re - im \cdot im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{-1 \cdot {im}^{2}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left({im}^{2}\right)} \]

      2. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{im \cdot im}\right) \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(im\right)\right) \cdot im} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(im\right)\right) \cdot im} \]

      5. lower-neg.f64 97.3

         \[\leadsto \color{blue}{\left(-im\right)} \cdot im \]

   5. Applied rewrites 97.3%

      \[\leadsto \color{blue}{\left(-im\right) \cdot im} \]

   if -5.0000000000022e-312 < (-.f64 (*.f64 re re) (*.f64 im im)) < +inf.0

   1. Initial program 100.0%

      \[re \cdot re - im \cdot im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{-1 \cdot {im}^{2}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left({im}^{2}\right)} \]

      2. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{im \cdot im}\right) \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(im\right)\right) \cdot im} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(im\right)\right) \cdot im} \]

      5. lower-neg.f64 11.0

         \[\leadsto \color{blue}{\left(-im\right)} \cdot im \]

   5. Applied rewrites 11.0%

      \[\leadsto \color{blue}{\left(-im\right) \cdot im} \]

   6. Taylor expanded in re around inf

      \[\leadsto \color{blue}{{re}^{2}} \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \color{blue}{re \cdot re} \]

      2. lower-*.f64 100.0

         \[\leadsto \color{blue}{re \cdot re} \]

   8. Applied rewrites 100.0%

      \[\leadsto \color{blue}{re \cdot re} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 54.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ re \cdot re \end{array} \]

FPCore

(FPCore re_sqr (re im) :precision binary64 (* re re))

C

double re_sqr(double re, double im) {
	return re * re;
}

Fortran

real(8) function re_sqr(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    re_sqr = re * re
end function

Java

public static double re_sqr(double re, double im) {
	return re * re;
}

Python

def re_sqr(re, im):
	return re * re

Julia

function re_sqr(re, im)
	return Float64(re * re)
end

MATLAB

function tmp = re_sqr(re, im)
	tmp = re * re;
end

Wolfram

re$95$sqr[re_, im_] := N[(re * re), $MachinePrecision]

Derivation

1. Initial program 95.7%

   \[re \cdot re - im \cdot im \]
2. Add Preprocessing

3. Taylor expanded in re around 0

   \[\leadsto \color{blue}{-1 \cdot {im}^{2}} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left({im}^{2}\right)} \]

   2. unpow2 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{im \cdot im}\right) \]

   3. distribute-lft-neg-in N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(im\right)\right) \cdot im} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(im\right)\right) \cdot im} \]

   5. lower-neg.f64 52.8

      \[\leadsto \color{blue}{\left(-im\right)} \cdot im \]

5. Applied rewrites 52.8%

   \[\leadsto \color{blue}{\left(-im\right) \cdot im} \]

6. Taylor expanded in re around inf

   \[\leadsto \color{blue}{{re}^{2}} \]
7. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \color{blue}{re \cdot re} \]

   2. lower-*.f64 53.8

      \[\leadsto \color{blue}{re \cdot re} \]

8. Applied rewrites 53.8%

   \[\leadsto \color{blue}{re \cdot re} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024288 
(FPCore re_sqr (re im)
  :name "math.square on complex, real part"
  :precision binary64
  (- (* re re) (* im im)))